Clifford algebra
Clifford algebra is a type of algebra in which the the geometric product is defined. Quick overview Complex numbers can be used to represent (and actually perform) rotations but only in 2 dimensions. Bivectors on the other hand can be used to represent rotations in any number of dimensions. But, to actually use a bivector to rotate a geometric object it's necessary to create an entirely new operation. This new operation is called the geometric product. The geometric product of a vector and a bivector results in a new vector which is equal to what the first vector would be if it were rotated around the axis of the bivector. : A(A\and B)=B Geometric product as higher dimensional equivalent of Complex numbers The geometric product is one way of generalizing the concept of complex numbers into higher dimensions. The geometric product is a multivector. In three dimensions a multivector is any sum of a scalar, vector, bivector, and a trivector. : M=\langle M\rangle_0+\langle M\rangle_1 +\langle M\rangle_2+\langle M\rangle_3 Geometric product of vectors A and B=AB=A\bullet B+A\and B : A\bullet B is the dot product of A and B which is a scalar. : A\and B is the wedge product of A and B which is a bivector. In three dimensions there exists a certain unit trivector ( e_1\and e_2\and e_3=e_{123}=I ) whose geometric product with itself is -1. (Multiplying by the equivalent of i converts anything, including itself, to its dual)Some books say to divide by but this only has the effect of changing the sign since 1/ = / 2 = /-1 = - . Therefore in three dimensions this unit trivector is the Clifford algebra equivalent of i. This is easier to understand if we first look at the two-dimensional case. In two dimensions a certain unit bivector would be the equivalent of i. A unit bivector represents a 90-degree turn so the square of a unit bivector would be a 180-degree turn. The geometric product of 2 vectors is a scalar plus a bivector and therefore in two Dimensions it is the clifford algebra equivalent of a complex number. In two dimensions AB=A\bullet B+A\and B=A\bullet B+I(A\times B)=\|a\|\|b\|\bigl\cos(\theta)+I\sin(\theta)\bigr=re^{\theta I} : A\times B is the 2-D cross product of A and B which is a scalar. Variables used e_1,e_2,e_3 are orthogonal unit vectors forming the standard basis and corresponding to axes x,y,z . A,B,C,D are unit vectors: : A=a_1e_1+a_2e_2+a_3e_3 : B=b_1e_1+b_2e_2+b_3e_3 : C=c_1e_1+c_2e_2+c_3e_3 : D=d_1e_1+d_2e_2+d_3e_3 u,v,w,x are arbitrary vectors. M,N are arbitrary multivectors. Definitions In mathematics, a Clifford algebra is an associative algebra (with a multiplicative identity) generated by a vector space with a quadratic form. Clifford algebra in 3 dimensions is called C 3. A bilinear form is a generalization of an inner product. : \beta(u,v)=\text{scalar} A bilinear form is symmetric if the order of the vectors does not matter. : \beta(u,v)=\beta(v,u) A bilinear form is Non-degenerate if it is true that whenever u\ne0,v\ne0 then \beta(u,v)\ne0 Quadratic form Q(x)=\beta(x,x) where \beta is a symmetric bilinear form. Q(x) is positive definite if: : \begin{cases}Q(x)>0&:x\ne0\\Q(x)=0&:x=0\end{cases} Polarization identity: In a normed space, if the parallelogram lawThe parallelogram law (also called the parallelogram identity) states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. For a rectangle the parallelogram law reduces to the Pythagorean theorem. holds, then there is an inner product on V that equals the square of the normWikipedia:Polarization identity : \|x\|^2=\langle x,x\rangle for all x\in V In Euclidean space: q(A)=B(A,A)=A\bullet A=\|A\|^2=a_1^2+a_2^2+a_3^2 Commutator product = C\times D:=\tfrac12(CD-DC) (Not to be confused with the cross product mentioned above) Formulas involving the standard basis in 3 dimensions Geometric product: : \begin{align} &e_1e_2=e_1\and e_2\\ &e_1(e_1e_2)=e_2\\ &e_1(e_2e_3)=e_1e_2e_3=e_{123}=I\\ &(e_1e_2)(e_2e_3)=e_1e_2e_2e_3=e_1e_3\\ &i=-e_2e_3\\ &j=-e_3e_1\\ &k=-e_1e_2\\ &i^2=j^2=k^2=ijk=-1\\ &I^2=e_1e_2e_3e_1e_2e_3=-1\\ &e_1I=e_1e_1e_2e_3=-e_1e_2e_1e_3=e_1e_2e_3e_1=Ie_1\\ &Ie_1=e_2e_3\\ &I^2e_1=Ie_2e_3=-e_1 \end{align} Formulas involving unit vectors in 3 dimensions Wedge product: : 3 \wedge 5 = 15 : A \wedge A = 0 : A \wedge B = bivector :: A \wedge B = (a_2 b_3 - a_3 b_2) e_23 + (a_3 b_1 - a_1 b_3)e_31 + (a_1 b_2 - a_2 b_1)e_12 : A \wedge B \wedge C = trivector :: A \wedge B \wedge C = (a_1 b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_1 b_3 c_2 - a_2 b_1 c_3 - a_3 b_2c_1)(e_1 \wedge e_2 \wedge e_3) : A \wedge B \wedge C \wedge D = 0 = a quadvector with zero volume : (A \wedge B \wedge C) \cdot C = A \wedge B Since A \wedge A = 0 : 0 = (A+B) \wedge (A+B) : 0 = A \wedge A + A \wedge B + B \wedge A + B \wedge B : 0 = 0 + A \wedge B + B \wedge A + 0 : 0 = A \wedge B + B \wedge A Therefore: : -(B \wedge A) = A \wedge B This means that rotation from B to A is the negative of rotation from A to B. Formulas involving bivectors in 3 dimensions For Bivectors (A \wedge B) and (C \wedge D) : : (A \wedge B)(C \wedge D) = (A \wedge B) \cdot (C \wedge D) + (A \wedge B) \times (C \wedge D) + (A \wedge B) \wedge (C \wedge D) Formulas involving antivectors in 3 dimensions :See also: Grassmann algebra \bar{e}_1, \bar{e}_2, \bar{e}_3 are pseudo-vectors or anti-vectors: : \bar{e}_1 = e_2 \wedge e_3 : \bar{e}_2 = e_3 \wedge e_1 : \bar{e}_3 = e_1 \wedge e_2 \bar{B} = b_1\bar{e_1} + b_2\bar{e_2} + b_3\bar{e_3} Wedge product of vector and antivector: : (a_1e_1 + a_2e_2 + a_3e_3) \wedge (b_1\bar{e}_1 + b_2\bar{e}_2 + b_3\bar{e}_3) = (a_1b_1 + a_2b_2 + a_3b_3)(e_1 \wedge e_2 \wedge e_3) = (A \cdot B)I A \cdot B = a_1b_1 + a_2b_2 + a_3b_3 AA = A \cdot A The Antiwedge product \vee operates on antivectors: : \bar{e}_1 \vee \bar{e}_2 = (e_2 \wedge e_3) \vee (e_3 \wedge e_1) = e_3 : \bar{e}_2 \vee \bar{e}_3 = (e_3 \wedge e_1) \vee (e_1 \wedge e_2) = e_1 : \bar{e}_3 \vee \bar{e}_1 = (e_1 \wedge e_2) \vee (e_2 \wedge e_3) = e_2 C \vee D := ((CI^{-1}) \wedge (DI^{-1}))I Formulas involving multivectors M \wedge N := \sum_{r,s}\langle \langle M \rangle_r \langle N \rangle_s \rangle_{r+s} Use in physics When the electromagnetic field is defined as the multivector sum of an electric field vector and a magnetic field bivector, the four Maxwell equations can be reduced to a single equation. Notes See also *Geometric algebra (A type of Clifford algebra limited to the reals) *symplectic Clifford algebra (Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms)Wikipedia:Weyl algebra References *Electromagnetism using Geometric Algebra versus Components *http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf *http://wiki.c2.com/?CliffordAlgebra *clifford-algebra-a-visual-introduction (uses an asterick to represent the geometric product) *http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/10/GA2015_Lecture2.pdf *Projective Geometry with Clifford Algebra - David Hestenes Category:Geometric_algebra